Construction of alpha-harmonic maps between spheres

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Universit`

a degli Studi di Pisa

Corso di Laurea Magistrale in Matematica

Tesi di laurea magistrale

Construction of α-harmonic maps

between spheres

Candidato Giada Franz

Relatore

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Introduction

Let (Mm_{, g) and (N}n_{, h) be two smooth compact Riemannian manifolds without}

boundary. Then we say that a function u ∈ W1,2(M, N ) is harmonic if it is a critical point for the Dirichlet functional, that is

E(u) := 1 2 ˆ M |∇u|2_dV M, where |∇u|2 _{= tr}

g(u∗h) is the Hilbert-Schmidt norm of ∇u.

Harmonic maps arise at the same time as generalizations of classical harmonic functions and closed geodesics in higher dimension. Indeed we recover the first ones in the case N = R and the second ones when M is 1-dimensional. Moreover, harmonic maps are deeply connected with minimal immersions. For example the two notions coincide in the case of isometric immersions.

Harmonic maps have been and are being studied widely in all their aspects (see for example [SY97], [EL78], [EL88] and [Str08] for a more complete overview), but for the purpose of this thesis we will treat only the case in which the dimension of M is 2. In fact this is a particularly interesting case to study, characterized by the conformal invariance of the Dirichlet energy.

Furthermore, harmonic maps in the 2-dimensional case enjoy good regular-ity properties. Actually a harmonic function u ∈ W1,2_{(M, N ) turns out to be}

C∞, while there are examples of harmonic functions everywhere discontinuous if dim M = 3, first found by Rivi`ere in [Riv95].

The natural question at this point is: do harmonic maps exist? Or, more precisely, given a function u0 ∈ W1,2(M, N ), does there exist a harmonic map

u ∈ W1,2_{(M, N ) homotopic to u} 0?

Let us list briefly what is known in this regard:

• The answer in general is no. For example there does not exist any harmonic map of degree ±1 from the 2-dimensional torus to the 2-dimensional sphere, as shown in [EW92].

• The problem has been solved affirmatively in [ES64] if the sectional curva-ture of N is non-positive. In fact this existence result holds without any assumption on the dimension of M .

• The answer is yes if dim M = 2 and π2(N ) = 0 ([Lem77] and [SU81]) or if

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Introduction

In the aforementioned work [SU81], Sacks and Uhlenbeck proved existence of harmonic maps in the following way: they introduced a perturbed Dirichlet energy and they showed existence of critical points of this functional, then they proved that critical points of these perturbed energies converge to a harmonic map.

In particular, given α ≥ 1 and u ∈ W1,2α(M, N ), they defined the α-energy of

u as Eα(u) := 1 2 ˆ M (2 + |∇u|2)αdVM

and they called α-harmonic maps the critical points of this functional. Notice that 1-harmonic maps are exactly the harmonic maps defined above.

The idea is that the Dirichlet energy suffers from a lack of compactness which makes existence hard to prove, whereas the perturbation with an exponent greater than 1 solves this problem. Indeed it is much easier to show existence of α-harmonic maps. On the other hand, it is useful to find α-harmonic maps since they converge (in a suitable sense) to harmonic maps when α goes to 1.

However, so far, the known α-harmonic maps with α ≥ 1 are all minimizers in their homotopy class. Hence, the aim of this thesis is to construct α-harmonic maps which do not fulfill this property and with an almost explicit description. In particular, we will try to construct this type of α-harmonic maps for each α > 1 sufficiently small when M = N = S2_.

Let us call mλ : S2 → S2 the maps given by the dilation of factor λ from C to

itself when seen through the stereographic projection from the north pole of the sphere. Then we will construct a 1-parameter family of maps uλ : S2 → S2 for

λ > 1 roughly defined as

• equal to m1/λ in a neighborhood of the north pole and equal to mλ in a

neighborhood of the south pole;

• equal to the symmetry with respect to the plane xy (looking to S2 _as

im-mersed in R3_{) elsewhere, with a proper gluing between this symmetry and}

the maps m1/λ and mλ in the contact area.

The construction of these maps is quite delicate and will require caution, but at the end we will manage to prove (up to technical hypotheses still to check) the following.

Theorem. For all α > 1 sufficiently small, there exists an α-harmonic map hα ∈

W1,2α_(S2_{, S}2_{) and λ}

α > 0 such that khα− uλαkW1,2 → 0 as α goes to 1. Moreover

λα → +∞ as α goes to 1.

Therefore we will obtain α-harmonic maps which look almost like the maps uλ.

In particular they will be homotopic to the identity but they will not minimize the

α-energy in their homotopy class.

The scheme to prove the theorem will be first to show existence of a critical point of the α-energy constrained to the family {uλ}λ>1and then to show that this

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gives rise to an α-harmonic map through a Lyapunov-Schmidt reduction, that is a procedure to reduce an infinite dimensional problem to a finite dimensional one. The rough heuristic idea to find the constrained critical point is that the maps

m1/λ and mλ we are trying to glue repulse each other (i.e. the glued map has

energy greater than the sum of the energy of the maps), while the perturbation of the energy functional tends to bring them closer. The work will consist in finding a balance between these two effects to obtain the sought critical point.

To summarize, the thesis will be structured as follows. We will start giving an overview of the classical theory of harmonic maps in Chapter 1. Then, in Chapter 2, we will introduce the perturbed functional of Sacks and Uhlenbeck and we will compute its first and second variation. Next, it will come the moment of presenting our problem and giving an idea of its proof in Chapter 3.

In Chapter 4 and 5 we will then enter in the details of the proof, first mak-ing some computations about dilations of the sphere and then facmak-ing the proof itself. Finally, we will conclude in Chapter 6 with the possible applications and generalizations of the problem.

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Notations

• (M, g) and (N, h) will always denote smooth compact Riemannian surfaces. • Often we will use h·, ·i in place of g, h (which one will be clear from the

context) to denote the metric tensor.

• ∇ denotes the covariant derivative on a Riemannian manifold induced from the Levi-Civita connection. Moreover we will denote _dt∇ the induced deriva-tive on curves.

• Γk

ij stands for the Christoffel symbols.

• R denotes the curvature tensor (both the (3, 1) and the (4, 0) versions of the tensor), where we will adopt the convention R(X, Y, Z, W ) = hR(Z, W )Y, Xi. • dV will denote the volume form on a Riemannian manifold.

• Given an immersion of a Riemannian manifold into another one, we will denote with A the second fundamental form and with H its trace, namely the mean curvature.

• W1,p _{will denote the Sobolev space of the function in L}p _{with first derivative}

in Lp _{(in particular we will use W}1,2 _{and W}1,2α_).

• ˆ_{C denotes the compactification C ∪ {+∞} of C.}

• λ will be a parameter going to +∞, while µ will often denote 1/λ.

• a(λ) ∼ b(λ) means a(λ) − b(λ) is o(a(λ)) and o(b(λ)) as λ → +∞; namely

a(λ) − b(λ) goes to 0 faster than both a(λ) and b(λ).

• a(λ) ≈ b(λ) means that a(λ) = O(b(λ)) and b(λ) = O(a(λ)) as λ → ∞; namely there exist two constants C1, C2 such that C1b(λ) ≤ a(λ) ≤ C2b(λ)

for λ sufficiently big.

• a(λ) b(λ) means that a(λ) = o(b(λ)) as λ → +∞.

• a(λ) / b(λ) means that a(λ) = O(b(λ)) as λ → +∞; namely there exists a constant C2 such that a(λ) ≤ C2b(λ) for λ sufficiently big.

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Chapter

1

Harmonic maps

In this chapter we want to give a brief introduction to the general theory of har-monic maps between manifolds, describing the main results in this regard and especially concentrating on the problem that brought to the introduction of a per-turbed functional. For a more complete overview on harmonic maps we refer to [SY97], [EL78], [EL88] and [Str08].

1.1 Definition

Let (Mm_{, g) and (N}n_{, h) be two smooth compact Riemannian manifolds without}

boundary. Thanks to the Nash embedding theorem (see [Nas56]), we can assume that N is isometrically embedded into some Rd _{for d ≥ 0.}

We will always denote with the indices i, j, k coordinates in M and with Greek letters α, β, γ coordinates in N .

Definition 1.1.1. We define the space W1,2_{(M, N ) as the space of functions u ∈}

W1,2_{(M, R}d_{) such that u(x) belongs to N for almost every x ∈ M .}

Definition 1.1.2. Given a function u ∈ W1,2_{(M, N ), we define its energy density}

as e(u) := 1₂|∇u|2_{, where}

|∇u|2 _{:= tr} g(u∗h)

is the Hilbert-Schmidt norm of ∇u.

Observe that, in local coordinates, the norm |∇u|2 _{is given by}

|∇u|2 _{= g}ij_(x) * ∂u ∂xi, ∂u ∂xj + N = gij(x) hαβ(u(x)) ∂uα ∂xi ∂uβ ∂xj .

Definition 1.1.3. Given a function u ∈ W1,2_{(M, N ), we define its Dirichlet energy}

as

E(u) :=

ˆ

M

e(u) dVM.

We say that the function u is harmonic if it is a critical point of the Dirichlet energy functional.

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Chapter 1. Harmonic maps

1.2 Euler-Lagrange equation

Considering variations ut ∈ W1,2(M, N ) with u0 = u, it turns out that the

Euler-Lagrange equation associated to the Dirichlet energy is given by ∆u + A(u)(∇u, ∇u) = 0 ,

where A is the second fundamental form of the immersion N ,→ Rd. Here by ∆u we mean the vector in Rd _{whose components are the Laplacian of the components}

of u and by A(u)(∇u, ∇u) we mean

A(u)(∇u, ∇u) := gijA(u) ∂u ∂xi,

∂u ∂xj

! .

We recall that the second fundamental form is defined as A(X, Y ) := ( ¯∇XY )⊥,

where ¯_{∇ is the covariant derivative in R}d_.

In the second chapter we are going to compute the Euler-Lagrange equation in a more general setting, thus we refer to that for a proof of this formula.

We now want to give an idea of what the Euler-Lagrange equation is telling us. In particular we are going to show some equivalent formulations. For the sake of simplicity, let us assume that the function u is C∞(M, N ).

First of all we have that ∆u = gij ∂ 2_u ∂xi_∂xj − g ij_Γk ij ∂u ∂xk = g ij_∇_¯ ∂u ∂xi ∂u ∂xj − g ij_Γk ij ∂u ∂xk ,

therefore the component of ∆u orthogonal to the tangent T N of N is (∆u)⊥ = gij ∇¯ ∂u ∂xi ∂u ∂xj !⊥ = gijA(u) ∂u ∂xi, ∂u ∂xj ! = A(u)(∇u, ∇u) .

Thus, the Euler-Lagrange equation writes simply as (∆u)> = 0, where (∆u)> is the component of ∆u along T N . Actually, it holds that

(∆u)> = ∆Nu ,

where ∆N_{u = tr}

M(∇Ndu) and the apex N denotes derivatives made on N . Hence,

we can say that being a harmonic map means that ∆N_{u = 0, accordingly to the}

classic notion of harmonicity.

Finally, notice that the Euler-Lagrange equation in coordinates (i, j, k in M and α, β, γ in N ) is ∆uγ+ gijΓγ_αβ∂u α ∂xi ∂uβ ∂xj = 0 ,

or, equivalently, we can write it as

gij∇∂u ∂xi ∂u ∂xj − g ij Γk_ij ∂u ∂xk = 0 . 2

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1.3. Examples

1.3 Examples

In this section we give a couple of basic examples of harmonic maps. In particular we see how harmonic maps and minimal immersions are connected in the basic case of isometric immersions.

Example 1.3.1. In the case N = R, the harmonic maps are the classical harmonic

maps, that is functions u : M → R such that ∆u = 0.

Example 1.3.2. If M = [0, 1], then the harmonic maps u : [0, 1] → N are

geodesics. Indeed the energy in this case is

E(u) = ˆ 1 0 ∂u ∂t 2 dt

and it is well-know that critical points of this functional are geodesics with constant speed parametrization.

Proposition 1.3.3. If u : M → N is an isometric immersion, then u is harmonic

if and only if it is a minimal immersion.

Proof. If we assume that u is an isometric immersion, then

∇∂u ∂xi ∂u ∂xj !>M = Γk_ij ∂u ∂xk,

where the apex >M denotes the component tangent to M . Moreover we have that

gij ∇∂u ∂xi ∂u ∂xj !⊥M = gijAM ∂u ∂xi, ∂u ∂xj ! = HM, where HM is the mean curvature of the immersion of M in N .

Therefore we have obtained that

gij∇∂u ∂xi ∂u ∂xj − g ij_Γk ij ∂u ∂xk = H M_,

which proves that u is harmonic if and only if the mean curvature of the immersion is 0, which is equivalent to being a minimal immersion.

1.4 Harmonic maps from surfaces

For our aims, we are particularly interested in the case in which m = dim M = 2. Therefore, from now on we will always implicitly assume this hypothesis if not explicitly said the opposite.

The key feature of this assumption is that the energy turns out to be confor-mally invariant. Hence, first of all let us check this property.

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Proposition 1.4.1. Let g1, g2 be two conformally equivalent metric on M . Then

Eg1_{(u) = E}g2_{(u) for all functions u ∈ W}1,2_{(M, N ).}

Clearly, with Eg _{here we intend the Dirichlet energy computed with respect to}

a metric g on M .

Proof. By definition of conformally equivalence, there exists f ∈ C∞(M ) such that

g2 = e2fg1. Then eg2(u) = e−2feg1(u) and

√

det g2 = e2f

√

det g1, where we have

used that dim M = 2 only in this last equality. Therefore we obtain also that dVg2

M = e2fdV g1

M and consequently we conclude

that Eg2_{(u) = E}g1_(u).

Remark 1.4.2. The conformal invariance of the energy functional in dimension 2

will be the cause of a lack of compactness of the functional itself. This will be the main reason behind the introduction of a perturbed functional, which will be central in the following.

Lemma 1.4.3. Let u : M → N be a (weakly) conformal immersion, then u is

harmonic if and only if it is a (branched) minimal immersion.

Proof. The proof follows easily from Propositions 1.3.3 and 1.4.1. Indeed, if u is

a conformal immersion, then it is an isometric immersion with respect to a metric on M conformally equivalent to the starting one. However, being harmonic is invariant by this change of metric and consequently we have reduced exactly to Proposition 1.3.3.

1.5 Regularity

In this section we want to get a glimpse on the regularity results for harmonic maps from surfaces. For the sake of completeness, we will also state the (negative) results in greater dimensions.

In fact, dimension 2 is special also for the regularity theory: this is the only case in which the harmonic maps are smooth. One of the reasons of this behavior is that W1,2_{(M, N ) enjoys good regularity properties if dim M = 2. An example}

is the following result, proven by Schoen and Uhlenbeck in [SU83].

Proposition 1.5.1. If dim M = 2, the space of smooth maps C∞(M, N ) is dense

in W1,2_{(M, N ).}

Remark 1.5.2. Always in [SU83], the two authors observed that the result is not

still true if dim M > 2.

A first consequence of Proposition 1.5.1 is that we can define the homotopy class of maps in W1,2_{(M, N ). This fact will be useful for existence results of harmonic}

maps in a given homotopy class and can be found in [Str08, 6.2 Theorem].

As anticipated, harmonic maps from surfaces are smooth; here we state the result, proved by H´elein in [H´el90].

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1.6. Existence results

Theorem 1.5.3. If dim M = 2 and u : M → N is harmonic, then u ∈ C∞(M, N ).

Remark 1.5.4. Already if dim M = 3, there exist examples of harmonic maps

which are everywhere discontinuous (see [Riv95]). However, in every dimension, a harmonic map u is smooth as soon as it is α-H¨older for some α ∈ (0, 1) (see [SY97]).

1.6 Existence results

We may now ask ourselves whether harmonic maps exist. In particular, one of the basic existence questions is the following: given a function u0 ∈ W1,2(M, N ), does

there exist a harmonic map u ∈ W1,2(M, N ) homotopic to u0?

The answer is not always positive, indeed for example there does not exist any harmonic map of degree ±1 from the 2-dimensional torus to the 2-dimensional sphere. The proof of this fact can be found in [EW92].

It turns out that the existence problem is easier (actually possible) to solve if the codomain manifold N has non-positive sectional curvature. In fact we have the following theorem, due to Eells and Sampson in [ES64].

Theorem 1.6.1 (Eells-Sampson). If the sectional curvature of N is non-positive,

then for every function u0 ∈ W1,2(M, N ) there exists a harmonic map u : M → N

homotopic to u0.

Remark 1.6.2. A first reason for this simpler behavior in the case of non-positive

scalar curvature can be understood looking to the second variation of the Dirichlet energy, which turns out to be positive definite in this case. The formula of the second variation can be found in the next chapter, where it is computed more in general for the α-energy we are going to define in the following.

Remark 1.6.3. Actually this theorem holds without the assumption on the

dimen-sion of M and it has been extended to the boundary case, as described for example in [SY97, Theorem 6.2].

Now we want to present a further existence result which holds only if dim M = 2. This theorem was obtained with different methods by Lemaire ([Lem77]) and by Sacks and Uhlenbeck ([SU81]). In particular, in order to prove this result, Sacks and Uhlenbeck introduced the perturbed functional we will talk about in the next chapter, for which existence is easier than for the Dirichlet energy. We will deepen this argument later, since this perturbed functional will be central in our work.

Theorem 1.6.4 (Lemaire, Sacks-Uhlenbeck). If dim M = 2 and π2(N ) = 0

(namely the second homotopy group of N is trivial), then for every function u0 ∈

W1,2_{(M, N ) there exists a harmonic map u : M → N homotopic to u} 0.

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1.7 Harmonic maps from the sphere

We want to conclude this chapter concentrating on harmonic maps from the 2-dimensional sphere: this will be in fact the setting of our problem.

First of all we prove that every harmonic map from the 2-dimensional sphere is weakly conformal. The proof is interesting by itself, since it give us a taste of the connection between harmonic maps from a surface and holomorphic maps.

Proposition 1.7.1. Let u : S2 _{→ N be a harmonic map, then u is a weakly}

conformal immersion.

Proof. Let us consider the holomorphic structure on S2 _{given by the stereographic}

projections. In any of these charts, the metric writes as λ(ζ) dζ ⊗ d¯ζ = λ(x + iy)(dx2 + dy2) for some conformal factor λ(ζ) > 0, where ζ = x + iy. Therefore, it is not difficult to see that a map u : S2 _{→ N is harmonic if and only if}

∇∂u ∂x ∂u ∂x + ∇∂u∂y ∂u ∂y = 0 ⇐⇒ ∇∂u∂ ¯ζ ∂u ∂ζ = 0 ,

which means that the term gij_Γk ij

∂u

∂xk is null.

Fixing one of these charts, let us now define the function ϕ as

ϕ(ζ) = * ∂u ∂ζ, ∂u ∂ζ + N . Note that this function is holomorphic in ζ, since

∂ϕ ∂ ¯ζ = 2 * ∇∂u ∂ ¯ζ ∂u ∂ζ, ∂u ∂ζ + N = 0 .

Moreover, let us define the form Φ := ϕ(ζ) dζ ⊗ dζ, which is in general called the Hopf differential 1_{. This form Φ is invariant by change of coordinates, hence}

it is a well-defined holomorphic 2-form in the whole S2_.

Consider now the change of charts given by ζ 7→ w = 1_ζ, in these new coordi-nates Φ writes as w−4_ϕ(1

w) dw ⊗ dw. However, we know that w −4_ϕ(1

w) must be

holomorphic in w = 0, thus ϕ(_w1) ≤ Cw4 _{for some C > 0. As a consequence, ϕ}

is holomorphic also at infinity and thus it has to be constant by Liouville’s the-orem. However, the inequality ϕ(1

w) ≤ Cw

4 _{tells that actually ϕ has to be zero}

everywhere.

Now observe that

ϕ(ζ) = 1 4 ∂u ∂x _N − ∂u ∂y _N ! + i 2 * ∂u ∂x, ∂u ∂y + N ,

1_{The Hopf differential can be defined for harmonic maps from a general surface domain and} it turns out to be a holomorphic 2-form in general.

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1.7. Harmonic maps from the sphere

therefore Φ = 0 on S2 _{if and only if u is weakly conformal and this concludes the}

proof.

Finally we mention a further existence result which holds in this case and which was obtain by Sacks and Uhlenbeck with the same methods of Theorem 1.6.4.

Theorem 1.7.2 (Sacks-Uhlenbeck). If M = S2 and the universal covering space of N is not contractible, then there exists a non-trivial harmonic map u : S2 _{→ N .}

Thanks to Lemma 1.4.3 and Proposition 1.7.1, as soon as we have existence of harmonic maps, we obtain existence of minimal immersions. Hence, in particular, we obtain the following corollary of the previous theorem.

Corollary 1.7.3. If M = S2 _{and the universal covering space of N is not}

con-tractible, then there exists a non-trivial smooth conformal branched minimal im-mersion u : S2 _{→ N .}

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Chapter

2

Perturbed functional

As anticipated in the previous chapter, we are now going to define and present the perturbed functional defined by Sacks and Uhlenbeck in [SU81]. In particular, we will first explain the motivations for the introduction of this functional, then we will set out its main properties (mainly compactness and convergence) and finally we will compute its first and second variation in detail.

2.1 Motivations

We have already anticipated the main reason for the introduction of a perturbed Dirichlet functional in Remark 1.4.2: the conformal invariance of the Dirichlet energy when the domain has dimension 2 (proved in Proposition 1.4.1) causes a lack of compactness, which makes the problem of existence hard to solve.

Let us clarify better this fact through an example. In particular let us consider the case in which domain and codomain are both equal to S2.

First of all we need the definition of M¨obius transformations of the sphere.

Definition 2.1.1. The M¨obius group (whose elements are called M¨obius trans-formations), denoted with M, is the set of holomorphic maps of degree 1 from

ˆ

C = C ∪ {+∞}∼= S2 to itself, which are of the form

ζ 7→ aζ + b cζ + d

with a, b, c, d ∈ C such that ad − bc = 1.

The M¨obius transformations are easily seen to be conformal transformations of the sphere. In fact, together with their complex conjugates, they represent all these conformal maps. Therefore, in this case, there are plenty of conformal transformations (in particular a non-compact set) and consequently there is no hope of applying a compactness argument to prove the existence of harmonic maps.

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Chapter 2. Perturbed functional

The idea of Sacks and Uhlenbeck was then to perturb the Dirichlet functional, breaking the conformal symmetry, in order to gain compactness. In this way they obtained existence of critical points of the perturbed functional easily and then they proved that these critical points converge to a harmonic map.

2.2 Definition

Having understood the reasons of Sacks and Uhlenbeck, we can now define the perturbed functional they introduced in [SU81]. Let us recall that we are assuming that M and N are smooth compact Riemannian manifold without boundary and dim M = 2, hypotheses which will be take for granted in the following.

Definition 2.2.1. For every α > 1 and for every u ∈ W1,2α_{(M, N )} 1_{, we define}

its α-energy as Eα(u) := 1 2 ˆ M (2 + |∇u|2)αdVM.

We say that a function u ∈ W1,2α_{(M, N ) is α-harmonic if it is a critical point for}

the perturbed functional Eα.

Remark 2.2.2. Observe that E1(u) =

´

M dVM + E(u) = |M | + E(u), thus the

1-harmonic maps coincide with the harmonic maps.

One may wonder why to choose this functional rather than, for example, the easier one 1₂´_M|∇u|2α_dV

M. The answer is that sometimes we need the uniform

ellipticity of the Euler-Lagrange equation, which is satisfied by this choice and not by the simpler one. Indeed the Euler-Lagrange equation of the easier functional contains |∇u| in the denominator, thus it is problematic for |∇u| near 0.

Remark 2.2.3. Actually, we have made a slight modification of the functional

de-fined by Sacks and Uhlenbeck, which was 1₂´_M(1 + |∇u|2)αdVM. However, this is

only a matter of convenience for the computations.

Before going on with the compactness and convergence properties of these perturbed functional, we want to mention the regularity of α-harmonic maps, as stated for harmonic maps in Theorem 1.5.3.

Theorem 2.2.4 ([SU81, Proposition 2.3]). If u ∈ W1,2_{(M, N ) is an α-harmonic}

map with α > 1, then u ∈ C∞(M, N ).

2.3 Compactness

We now want to specify in which sense we “gain compactness” perturbing the Dirichlet functional. Thus, first of all, let us define the compactness condition

1_{Here the definition of W}1,2α_{(M, N ) is completely analogous to Definition 1.1.1.}

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2.4. Convergence

which fits our case, that is the Palais-Smale condition, and the compactness the-orem which follows from this condition.

Definition 2.3.1. We say that a C1 _{function I : X → R from a Banach manifold}

X satisfies the Palais-Smale condition if every subset S ⊆ X on which I is bounded

and kdI(·)k is not bounded away from 0 has closure ¯S which contains a critical

point of I.

Theorem 2.3.2 ([Pal68]). If I is a C2_{-function on a complete separable C}2_Finsler

manifold and it satisfies the Palais-Smale condition with respect to the Finsler structure, then I takes on its minimum in each component.

Finally, as anticipated, we state here that our functional Eα actually fulfills the

Palais-Smale condition.

Proposition 2.3.3 ([Pal68]). The functional Eα is C2 on the Banach manifold

W1,2α_{(M, N ) and satisfies the Palais-Smale condition in a complete Finsler metric}

on the manifold W1,2α(M, N ).

Notice that this fact, thanks to Proposition Theorem 2.3.2, gives easily exis-tence of α-harmonic maps in every connected component of W1,2α(M, N ). Though, some more work is needed to obtain non-trivial (that is non-constant) α-harmonic maps.

However, we are not interested in looking into the details of the work by Sacks and Uhlenbeck and we settle for giving just an idea for the proof of existence.

2.4 Convergence

In this section we just want to set out the convergence results carry out by Sacks and Uhlenbeck.

First of all let us notice that, given a sequence {uα}α>1 of α-harmonic maps

with bounded α-energy, there exists a subsequence which weakly converges in

W1,2(M, N ) to a map u ∈ W1,2(M, N ).

Then, Sacks and Uhlenbeck first proved that this subsequence actually con-verges (up to extract a further subsequence) in C1 _{except that in a finite number}

of points, as clarified in the following theorem.

Theorem 2.4.1. Let {uα}α>1 be a sequence of harmonic maps with bounded

α-energy which weakly converges to u in W1,2_{(M, N ). Then there exists a subsequence}

and a finite number of points {x1, . . . , xl} such that this subsequence converges in

C1_{(M \ {x}

1, . . . , xl}, N ).

Finally, Sacks and Uhlenbeck described what happens in the points in which there is not C1 _convergence.

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Theorem 2.4.2. Let {uα}α>1 be a sequence of α-harmonic maps with bounded

α-energy such that uα → u in C1(M \ {x1, . . . , xl}, N ), but such that there is not

convergence in C1_{(M \ {x}

2, . . . , xl}, N ). Then there exists a non-trivial harmonic

map ˜u : S2 _{→ N such that}

˜ u(S2) ⊆ \ m→∞ [ α→1 \ β≤α uβ(x1, 2−m) .

Moreover E(u) + E(˜u) ≤ lim sup_α→1E(uα).

Remark 2.4.3. Actually, we will see this kind of blow up around a point for the α-harmonic maps we are going to construct.

Remark 2.4.4. In [Lam10], Lamm proved a better quantization for the energy of

the limit. Indeed he showed that there is no energy loss assuming a further entropy condition.

One can wonder if all harmonic maps can be approximated by α-harmonic maps. However, Lamm, Malchiodi and Micallef in [LMM15] proved that this is not true, showing a counterexample in the case in which domain and codomain are equal to S2_.

2.5 First variation

In these last two sections we want to compute the first and the second variations of the α-energy. In order to simplify the notation, we will omit the subscript N used in the first chapter to specify that we were making norms and scalar products with respect to the metric on N . Moreover, we will often use ∂i to denote _∂x∂i.

Proposition 2.5.1. Let X ∈ C∞(M, T N ) be such that X(x) ∈ Tu(x)N for almost

every x ∈ M . Then we have that

dEα(u)[X] = − ˆ M hτα(u), Xi dVM, where τα(u) := α(2 + |∇u|2)α−1 " ∆u + 2(α − 1) 2 + |∇u|2 D ∇2_{u, ∇u}E_∇u #

is called the α-tension field of u.

Here with ∆u we mean ∆u := tr(∇du) = gij(∇du)γ_ij∂γ and 2 h∇2u, ∇ui ∇u is

a notation for 2gij_h∇

∂iu∇u, ∇ui ∂ju = g

ij_∂

i|∇u|2∂ju.

Remark 2.5.2. Notice that τα(u) = 0 is the Euler-Lagrange equation associated

to the α-energy Eα. In addition, observe that for α = 1 we obtain exactly the

Euler-Lagrange equation described in Section 1.2.

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2.5. First variation

Proof. Consider a variation ut of u such that ∂u_∂tt

_t=0 = X. An example of such a variation is ut(x) = expu(x)(tX(x)). Then the first variation of the α-energy Eα at

u along X is given by dEα(u)[X] = dE(ut) dt _t=0,

so let us compute it. In the following, we will take for granted that all the deriva-tives are computed for t = 0. Then we have that

dE(ut) dt _t=0 = d dt 1 2 ˆ M (2 + |∇ut|2)αdVM = 1 2 ˆ M α(2 + |∇u|2)α−1 d dt gijh∂iut, ∂juti dVM = ˆ M α(2 + |∇u|2)α−1gij ∇ dt ∂iut, ∂ju dVM = ˆ M α(2 + |∇u|2)α−1gij * ∇∂iu ∂ut ∂t , ∂ju + dVM = ˆ M α(2 + |∇u|2)α−1gij(∂ihX, ∂jui − hX, ∇∂iu∂jui) dVM.

Integrating by parts the first term, we obtain that − ˆ M α(2 + |∇u|2)α−1gij∂ihX, ∂jui dVM = = ˆ M α(α − 1)(2 + |∇u|2)α−2∂i|∇u|2gijhX, ∂jui dVM + + ˆ M α(2 + |∇u|2)α−1∂igijhX, ∂jui dVM + + ˆ M α(2 + |∇u|2)α−1 1 2 g ij_gkl_∂ igklhX, ∂jui dVM,

where we have used that ∂idVM = 1₂gkl∂igkldVM. Therefore, recalling that

(∇du)γ_ij = (∇∂iu∂ju) γ_{− Γ}k ij∂kuγ = (∇∂iu∂ju) γ_{+ ∂} igij∂juγ+ 1 2g ij_gkl_∂ igkl∂juγ, we conclude that dEα(u)[X] = − ˆ M α(2 + |∇u|2)α−1 * ∆u + 2(α − 1) 2 + |∇u|2 D ∇2_{u, ∇u}E_{∇u, X} + dVM,

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2.6 Second variation

We are now interested in computing the second variation of the α-energy Eα.

Proposition 2.6.1. Let X, Y ∈ C∞(M, T N ) be such that X(x), Y (x) ∈ Tu(x)N

for almost every x ∈ M . Then we have that

d2Eα(u)[X, Y ] = = ˆ M α(2 + |∇u|2)α−1hgijD∇∂iuX, ∇∂juY E − gij_R(∂ iu, X, ∂ju, Y ) i dVM + + 2 ˆ M α(α − 1)(2 + |∇u|2)α−2gijh∇∂iuX, ∂jui g kl_h∇ ∂kuY, ∂lui dVM.

Here, for the curvature tensor R, we adopt the definition R(X, Y, Z, W ) =

hR(Z, W )Y, Xi.

Proof. Let ut,s be a variation of u such that

∂ut,s ∂t _t,s=0 = X and ∂ut,s ∂s _t,s=0 = Y . Moreover we require that, calling Xs=

∂ut,s ∂t _t=0, it holds ∇ dsXs= 0.

Then the second variation of Eα at u along X, Y is given by

d2Eα(u)[X, Y ] = ∂2_E α(ut,s) ∂t∂s _t,s=0 = ∂ ∂s _s=0 ∂Eα(ut,s) ∂t _t=0 = ∂ ∂s _s=0dEα(u0,s)[X] . For convenience, we will write us instead of u0,s and we will omit the evaluation

in s = 0 of the derivative, therefore we have that d2Eα(u)[X, Y ] = ∂ ∂sdEα(us)[X] = − ˆ M ∇ dsτα(us), X dVM.

Hence, let us compute _ds∇τα(us); we have that

∇ dsτα(us) = ∇ ds " α(2 + |∇us|2)α−1 " ∆us+ 2(α − 1) 2 + |∇us|2 D ∇2us, ∇us E ∇us ## = α(α − 1)(2 + |∇u|2)α−2 d|∇us| 2 ds " ∆u + 2(α − 1) 2 + |∇u|2 D ∇2_{u, ∇u}E ∇u # + + α(2 + |∇u|2)α−1 " ∇ ds(∆us) + (α − 1)g ij∇ ds ∂i|∇us|2∂jus 2 + |∇us|2 !# .

Now let us compute the three derivatives in s in the last expression separately. The first one is equal to

∂|∇us|2 ∂s = ∂ ∂s(g kl_h∂ kus, ∂lusi) = 2gkl ∇ ds∂kus, ∂lu = 2gkl * ∇∂ku ∂us ∂s, ∂lu + = 2gklh∇∂kuY, ∂lui . 14

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2.6. Second variation

Then let us deal with the derivative of ∆us, which turns out to be

∇ ds(∆us) = ∇ ds gij∇∂ius∂jus− g ij_Γk ij∂kus = gij∇ ds∇∂ius∂jus− g ij_Γk ij ∇ ds∂kus = gij∇∂iu ∇ ds∂jus+ g ij R ∂us ∂s , ∂iu ! ∂ju − gijΓkij∇∂ku ∂us ∂s = gij∇∂iu∇∂ju ∂us ∂s + g ij_{R (Y, ∂} iu) ∂ju − gijΓkij∇∂kuY = gij∇∂iu∇∂juY + g ij R (Y, ∂iu) ∂ju − gijΓkij∇∂kuY ,

notice that here we have used the definition of the curvature tensor R(X, Y )Z = ∇X∇YZ − ∇Y∇XZ − ∇[X,Y ]Z. Finally let us compute the last derivative

∇ ds ∂i|∇us|2∂jus 2 + |∇us|2 ! = = ∂ju 2 + |∇u|2 ∂ ∂s(∂i|∇us| 2_{) +} ∂i|∇u|2 2 + |∇u|2 ∇ ds∂jus− ∂i|∇u|2∂ju (2 + |∇u|2₎2 ∂ ∂s|∇us| 2 = ∂ju 2 + |∇u|2 ∂i ∂ ∂s|∇us| 2 ! + ∂i|∇u| 2 2 + |∇u|2 ∇∂ju ∂us ∂s − ∂i|∇u|2∂ju (2 + |∇u|2₎2 ∂ ∂s|∇us| 2 = ∂ju 2 + |∇u|2 ∂i 2gklh∇∂kuY, ∂lui + ∂i|∇u| 2 2 + |∇u|2 ∇∂juY + − ∂i|∇u| 2_∂ ju (2 + |∇u|2₎2 2gklh∇∂kuY, ∂lui .

We can now put all the computations together to obtain that ∇ dsτα(us) = α(2 + |∇u| 2₎α−1n_{(∆Y + g}ij_{R(Y, ∂} iu)∂ju)+ + 2(α − 1) 2 + |∇u|2 " gklh∇∂kuY, ∂lui ∆u + α − 1 2 + |∇u|2g ij_∂ i|∇u|2∂ju ! + + gij∂ju∂i(gklh∇∂kuY, ∂lui) + 1 2g ij_∂ i|∇u|2∇∂juY + − gij∂i|∇u| 2_∂ ju 2 + |∇u|2 g kl_h∇ ∂kuY, ∂lui #)

= α(2 + |∇u|2)α−1n(∆Y + gijR(Y, ∂iu)∂ju) +

+ 2(α − 1) 2 + |∇u|2 " gklh∇∂kuY, ∂lui ∆u + α − 2 2 + |∇u|2g ij_∂ i|∇u|2∂ju ! + + gij∂ju∂i(gklh∇∂kuY, ∂lui) + 1 2g ij ∂i|∇u|2∇∂juY .

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Now, integrating by parts the first term, we obtain the following − ˆ M α(2+|∇u|2)α−1h∆Y, Xi dVM = ˆ M α(2 + |∇u|2)α−1gijD∇∂iuX, ∇∂juY E dVM + + ˆ M α∂i h (2 + |∇u|2)α−1igijD∇∂juY, X E dVM = ˆ M α(2 + |∇u|2)α−1gijD∇∂iuX, ∇∂juY E dVM + + ˆ M α(α − 1)(2 + |∇u|2)α−2gij∂i|∇u|2 D ∇∂juY, X E dVM.

Notice the the second term here cancels with the last term in the computation of

∇

dsτα(us).

We now want to integrate by parts the other term in which a second derivative of Y appears and we obtain

ˆ M (2 + |∇u|2)α−2gij∂i(gklh∇∂kuY, ∂lui) h∂ju, Xi dVM = = ˆ M (2 + |∇u|2)α−2gklh∇∂kuY, ∂lui h∆u, Xi dVM + + (α − 2) ˆ M (2 + |∇u|2)α−3gklh∇∂kuY, ∂lui g ij_∂ i|∇u|2h∂ju, Xi dVM + + ˆ M (2 + |∇u|2)α−2gijh∇∂iuX, ∂jui g kl_h∇ ∂kuY, ∂lui dVM.

Now, collecting all the computations, we obtain exactly the desired formula.

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Chapter

3

Problem and tools

In this chapter, it is finally the moment to describe the problem we are going to face. Therefore, we will first specialize to the case M = N = S2 _{and we will give}

some basic definitions; then we will state our problem, trying to give an idea of its content and its proof.

3.1 Harmonic maps between spheres

Before stating the problem, we need some preliminary definitions and to familiarize with the new setting. Thus, in particular, we need to specialize the problem to the case in which M = N = S2 _{with its standard metric.}

Remark 3.1.1. From now on, we will implicitly identify S2 _{with ˆ}

C = C ∪ {+∞} through the stereographic projection from the north pole.

Observe that the stereographic projection is a conformal map, therefore it will not be a problem to think about maps from ˆC to S2 _{instead of maps from S}2 _to

S2_{, thanks to Proposition 1.4.1.}

First notice that, in this case, we have A(u) = u h·, ·i and R(X, Y, Z, W ) = hX, Zi hY, W i − hX, W i hY, Zi, which are terms appearing respectively in the first and second variation of the α-energy. Moreover, in this case it holds that E1(u) =

E(u) + 4π for every u ∈ W1,2(S2_{, S}2_).

Before going on, let us recall the definition of M¨obius transformation, already given in the previous chapter.

Definition 3.1.2. The M¨obius group (whose elements are called M¨obius trans-formations), denoted with M, is the set of holomorphic maps of degree 1 from

ˆ

C = C ∪ {+∞}∼= S2 to itself, which are of the form

ζ 7→ aζ + b cζ + d

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Chapter 3. Problem and tools

Remark 3.1.3. Recall that the M¨obius transformations and their complex conju-gates are all and only the conformal maps from S2 to S2.

We now want to emphasize that this case is a particularly nice to study. Indeed the harmonic maps from S2 _{to S}2 _{are completely characterized, as the following}

theorem by Wood and Lemaire describes.

Theorem 3.1.4 ([EL78, (11.5)]). The harmonic maps between 2-spheres are

pre-cisely the rational maps and their complex conjugates (that is rational maps in ζ or ¯ζ) on ˆ_{C = C ∪ {+∞}}∼= S2.

In particular, the M¨obius transformations together with their complex conju-gates constitute all the harmonic maps of degree 1 from S2 _{to S}2_.

Actually, in our problem, we will deal only with maps of degree 1. Therefore from now on we will consider only this case.

Finally, let us conclude the section proving a lower bound for the α-energy, specialized to the case of maps of degree 1. The proposition will characterize also all the minimizers of the α-energy between the maps of degree 1.

Proposition 3.1.5. Given α ≥ 1 and a map u ∈ W1,2α_{(M, N ) of degree 1, it}

holds that Eα(u) ≥ 22α+1π.

Moreover, equality holds if and only if u is a conformal transformation if α = 1 and if and only if u is a conformal transformation with constant energy density equal to 2 if α > 1.

Proof. Recall (see e.g. [Ham99]) that the degree of u can be defined as

deg(u) = 1 4π ˆ S2 u∗( dVS2) = 1 4π ˆ S2 J (u) dVS2,

where J (u) = u · e1(u) ∧ e2(u), with {e1, e2} local oriented orthonormal basis of

T S2_{, is the Jacobian of u. Therefore, if deg(u) = 1, we obtain}

8π = ˆ S2 1 + J (u) dVS2 ≤ ˆ S2 1 + 1 2|∇u| 2_dV S2 ≤ (21−αE_α(u)) 1 α(4π) α−1 α ,

from which follows that Eα(u) ≥ 22α+1π.

Notice that the first inequality holds with equality if and only if u is a conformal transformation. On the other hand, if α > 1, the last H¨older inequality holds with equality if and only if u has constant energy density, which must be in particular equal to 2.

Remark 3.1.6. The previous proposition tells us that all the M¨obius transforma-tions have Dirichlet energy equal to 4π or, analogously, 1-energy equal to 8π.

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3.2. Dilations

3.2 Dilations

Consider a matrix M = a b c d ! ∈ SL(2, C) = {A ∈ Mat(2, C) : det A = 1}

which represents a generic M¨obius transformation. Thanks to the singular value decomposition, there exist U, V ∈ SU(2, C) = {A ∈ SL(2, C) : AA∗ = A∗_{A = 1}} such that M = U DV∗, where

D = λ 1 2 0 0 λ−12 ! .

Therefore, up to rotations, all the M¨obius transformations are dilations, that we define here below.

Definition 3.2.1. We will call dilations the elements of the M¨obius group given by mλ(ζ) := λζ with λ > 0 a real number. Notice that they correspond to M¨obius

transformations with a = λ12, d = λ− 1

2 and b = c = 0.

How do the dilations look on the sphere? Figure 3.1 shows the definition of a dilation (of factor λ > 1) making explicit the step through the stereographic projection. C S2 N p ζ λζ mλ(p)

Figure 3.1: Definition of dilations seen in section.

Observe that, if λ < 1, a dilation squeezes the sphere to the south pole along the meridians. The behavior for λ > 1 is exactly the same but the sphere is sent from the south pole to the north pole.

Moreover notice that, if λ > 1 is sufficiently big, almost all the Dirichlet energy of the dilation mλ will be concentrated near the south pole, as we will clarify in

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Chapter 3. Problem and tools N S mλ for λ < 1 N S mλ for λ > 1

Figure 3.2: Behavior of dilations.

3.3 Representation in polar coordinates

It will be useful in the following to represent the sphere in polar coordinates, thus let us briefly fix some notation. The polar parametrization of S2 _{is given by}

[0, π]×[0, 2π] → S2 _{⊆ R}3

(r, θ) 7→ (sin r cos θ, sin r sin θ, cos r) . In these coordinates the stereographic projection π : S2 _{→ ˆ}

C is given by

π(r, θ) = sin r

1 − cos r e

iθ

.

Every rotationally symmetric function ug : S2 → S2 in polar coordinates can

be written as

ug(r, θ) = (g(r), θ) = (sin(g(r)) cos θ, sin(g(r)) sin θ, cos(g(r))) ,

for some g : [0, π] → R, and its energy density is equal to

e(ug) = 1 2 (g 0 )2 +sin 2_(g(r)) sin2_r ! . Indeed it holds that

∂ug ∂r = |g0(r)| and ∂ug ∂θ = |sin r| .

3.4 Statement

As the title suggest, the aim of this thesis is to construct α-harmonic maps from S2

to itself. In particular, these α-harmonic maps will not be conformal maps, neither

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3.4. Statement

they will minimize the α-energy in their homotopy class (as all the α-harmonic maps constructed hereto do). Moreover it will be possible to provide an almost explicitly description of them.

The rough idea from which we started is the following: trying to glue two opposite dilations (one of factor λ and the other of factor 1/λ) in order to find a repulsion between the two. The repulsion means that the Dirichlet energy of the gluing is decreasing as λ goes to +∞. This will be an interesting behavior because, as we will see in the following, the α-energy tends to bring together the two dilations. Thus the hope is to find a balance between these two effects in order to give rise to a critical point.

Obviously this is pure speculation, thus let us begin to say something formally. First of all, we are going to construct a 1-parameter family of maps {uλ}λ>1

from S2 _{to itself, which are the “gluing” mentioned above and which will play the}

role of pseudo-critical points.

Let us approximately describe the form of uλ. In particular, it will be:

• a rotationally and equatorially symmetric map, namely a map invariant for rotations around the axis through the poles and for the symmetry with re-spect to the plane xy, which splits the sphere in the upper and lower semi-sphere;

• equal to m1/λ in a neighborhood of the north pole and equal to mλ in a

neighborhood of the south pole;

• equal to the symmetry with respect to the plane xy in the middle of the two regions, with a proper gluing between this symmetry and the dilations.

N S m1/λ mλ symmetry w.r.t. the plane xy uλ interaction zone

Figure 3.3: Idea of the definition of the gluing uλ.

Having given an idea about the setting, let us formally define the family {uλ}λ>1. The attachment between the dilations and the symmetry may look quite

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complicated, anyhow we will try to explain why is constructed in this way in the next section.

Definition 3.4.1. Consider smooth functions 0 < _λ1 < r1(λ) < r2(λ) < r3(λ) <

r4(λ) < π₂ depending on λ (from now on we will omit this dependence from λ of

the functions ri(λ) with i = 1, 2, 3, 4) with the following behavior for λ going to

+∞: • r2

1 ∼ 1λ (notice in particular that 1 λ  r1); • 1 r2 r1 log _r 3 r2 ∼ 1 4 log λ; • r4 ≥ 2r3.

Then define uλ ∈ W1,2(S2, S2) in polar coordinates as

uλ(r, θ) := (gλ(r), θ) , where gλ ∈ W1,2([0, π], [0, 3π]) is given by gλ(r) :=                      f1/λ(r) for 0 ≤ r ≤ r1

arcsinsin(f1/λ(r)) + sin

π − r−r1 r2−r1r2 ∈ (π 2, 3π 2 ) for r1 ≤ r ≤ r2

arcsinsin(f1/λ(r)) + sin(π − r)

∈ (π 2, 3π 2 ) for r2 ≤ r ≤ r3 arcsinr4−r r4−r3 sin(f1/λ(r)) + sin(π − r) ∈ (π 2, 3π 2 ) for r3 ≤ r ≤ r4 s(r) = π − r for r4 ≤ r ≤ π₂

and analogously for π₂ ≤ r ≤ π.

Remark 3.4.2. In fact uλ belongs to W1,2α(S2, S2) for all α ≥ 1.

Remark 3.4.3. One possibility for the choice of r1, r2, r3, r4 which fulfills all the

requirements is

r1 = λ−1/2, r2 = λ−1/2log(log λ) ,

r3 = λ−1/4log(log λ) , r4 = 2λ−1/4log(log λ) .

Indeed, in this way, r2 1 = 1 λ, r4 = 2r3 and finally 1 r2 r1 = log(log λ) log _r 3 r2 = 1 4log λ . At this point, we can state our theorem.

Theorem 3.4.4. For all α > 1 sufficiently small, there exists an α-harmonic map

hα ∈ W1,2α(S2, S2) and ηα > 0 such that khα − uηαkW1,2 → 0 as α goes to 1.

Moreover ηα→ +∞ as α goes to 1.

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3.5. Idea of the proof

Remark 3.4.5. Observe that, thanks to the closeness of these α-harmonic maps to

the family {uλ}λ>1, we will obtain that

1. hα is homotopic to the identity on the sphere;

2. E(hα) → 3E(1S2) = 12π as α goes to 1 and E_α(h_α) > 22α+1π, thus h_α is

easily not a minimizer of the α-energy in its homotopy class for α sufficiently close to 1;

3. hα → s in C1(S2\ {N, S}) as α → 1, where N and S are the north and the

south pole respectively. Moreover near the north and the south pole we have the blow up described by Theorem 2.4.2.

3.5 Idea of the proof

The idea for the proof of the theorem is first to find a critical point of the α-energy constrained to the family {uλ}λ>1for all α > 1 sufficiently small and then to prove

that, up to move slightly from this point, we find an α-harmonic map. Thus let us see separately how to deal with the two parts.

Remark 3.5.1. For further study of the same method we refer to [AM06], where

both the abstract setting and several applications (for example to the nonlinear Schr¨odinger equation) can be found.

Find a constrained critical point

The very rough idea has been described in the previous section: we hope that the gluings uλ that we have defined are such that their Dirichlet energy is decreasing

as λ goes to +∞. This is because the α-energy of a dilation mλ is increasing for

λ going to +∞ and therefore we hope that, thanks to the combination of these

two effects, the α-energy of uλ is convex when varying λ. Figure 3.4 shows the

behavior we are going to see.

λ E(uλ)

Eα(mλ)

Eα(uλ)

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This will be possible to prove only because of the accurate definition of the interaction between the dilations and the symmetry in the mid part of the sphere. Indeed let us look to what happens if we define the gluing in the most obvious way, that is ˜ gλ(r) :=        f1/λ(r) for 0 ≤ r ≤ ˜r s(r) = π − r for ˜r ≤ r ≤ π − ˜r fλ(r) for π − ˜r ≤ r ≤ π

where ˜r = arccosλ−1_λ+1 is chosen in such a way that ˜gλ is continuous. Then the

Dirichlet energy of ˜uλ(r, θ) := (˜gλ(r), θ) is equal to

E(˜uλ) = 4π 3 − 4 λ + 1 ,

which is increasing for λ large, as shown in Figure 3.5. Therefore there is no hope to obtain a critical point of Eα constrained to the family {˜uλ}λ>1.

λ E(˜uλ)

Figure 3.5: Behavior of the α-energy of the trivial gluing.

Then, which is the right way to define the gluing? The model to follow are the standing waves of the nonlinear Schr¨odinger equation, that is solutions u ∈

W1,2_(Rn_{) to the equation}    −∆u + u = up u > 0 in Rn_{, where 1 < p <} n+2

n−2. Indeed, in this case, the sum of a positive and a

negative standing wave has decreasing energy as the distance of the two picks goes to +∞ (namely a positive and a negative solution repulse each other).

It can be seen that the behavior of sin(fλ(r)) where 1_λ  r 1 is similar to that

of the standing waves away from their pick. Therefore this heuristic suggests to sum these components to make the maps interacting. Indeed this is exactly what we have done in the strip r2 ≤ r ≤ r3. The zones r1 ≤ r ≤ r2 and r3 ≤ r ≤ r4 are

there just to make the map in W1,2α_(S2_{, S}2_).

From the constrained critical point to the α-harmonic map

The tool to obtain an α-harmonic map from the constrained critical point is a Lyapunov-Schmidt reduction. Substantially the idea is the following.

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3.6. Lyapunov-Schmidt reduction

Let I : X → R be a functional defined on the Hilbert space X and let Z ⊆ X be a manifold of pseudo-critical points for I, which fulfills some hypotheses we are going to specify later. We want to see how a critical point of I restricted to Z gives rise to a critical point of I.

The first step is to find a manifold ˜Z = {z + w(z) : z ∈ Z} (called natural constraint) such that ∇I(z + w(z)) ∈ TzZ, with w(z) very small.

Then, instead of having a critical point of I restricted to Z we want a critical point ¯z of I(z + w(z)) restricted to Z, which in general will be found similarly.

However, given such ¯z, we have that ∇I(z + w(z)) is almost perpendicular to TzZ

(if ˜Z is sufficiently close to Z), but this is not possible unless ∇I(¯z + w(¯z)) is zero.

Z ˜ Z w(z) z ∇I(z + w(z))

The name “natural constraint” comes then from the fact that it is sufficient to find a critical point of I on ˜Z to find a critical point of I.

3.6 Lyapunov-Schmidt reduction

In this section we want to formalize the Lyapunov-Schmidt reduction, roughly explained in the previous section. There we talked about Hilbert spaces, but for our aims we will need to generalize the setting to Banach spaces.

So let us start with a preliminary invertibility lemma.

Lemma 3.6.1. Let (W1, k·k1) and (W2, k·k2) be Banach spaces and consider a C1

function F : W1 → W2. Assume that there exist δ, C, r > 0 such that

(PC) kF (0)k2 ≤ δ;

(INV) dF (0) is invertible with k(dF (0))−1k ≤ C;

(B) kdF (w) − dF (0)k ≤ 1

4C for every w ∈ W with kwk1 ≤ 2r.

Then, if Cδ ≤ r₂, there exists w ∈ W1 such that F (w) = 0 and kwk1 ≤ 2Cδ.

Moreover this solution of F (w) = 0 is unique in the ball of radius r.

Proof. The idea to find the sought zero is to use the contraction mapping theorem,

therefore first of all let us write the condition F (w) = 0 as a fixed point equation. In particular we have

F (w) = 0 ⇐⇒ −dF (0)[w] = F (0) + F (w) − F (0) − dF (0)[w]

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For w1, w2 ∈ ¯Br(0) we have that

kT (w1) − T (w2)k = k−(dF (0))−1(F (w1) − F (w2) − dF (0)[w1− w2])k ≤ CkF (w1) − F (w2) − dF (0)[w1− w2]k = C ˆ 1 0 (dF ((1 − s)w2+ sw1) − dF (0))[w1− w2] ds ≤ 1 4kw1− w2k ,

where we have used (B), and therefore T is a 1₄-contraction. Moreover it holds that kT (0)k ≤ Cδ and notice that, if kT (0)k ≤ Cδ ≤ r₂, then T maps the ball

¯

Br(0) into itself.

Consequently, by the contraction mapping theorem, T admits a fixed point w in ¯Br(0). In fact, T sends also the ball of radius 2Cδ into itself, therefore we obtain

a solution of F (w) = 0 with kwk ≤ 2Cδ.

We can now face the problem of finding a natural constraint for a functional given a manifold of approximate critical points.

Proposition 3.6.2. Let X be an affine Banach space. Consider a C2 _functional

I : X → R and a C1 _{manifold Z of finite dimension, which will play the role of}

a manifold of approximate critical points. Moreover suppose that for all z ∈ Z there exists Wz complementary to TzZ which has a C1 dependence on z ∈ Z. In

particular call Qz : TzX → TzZ and Pz : TzX → Wz the projections on the two

addends of the sum TzX = TzZ ⊕ Wz.

Assume that there exist δ, C, ˜_{C > 0 and a modulus of continuity ω : R}+ _{→ R}+

such that

(P C) kdI(z)k ≤ δ for all z ∈ Z;

(N D) kd2_{I(z)[w, ·]k ≥ Ckwk for all z ∈ Z and w ∈ W} z;

(B1) kd2I(z + w) − d2I(z)k ≤ ω(kwk) for all z ∈ Z and w ∈ Wz.

(B2) kd2I(z)[v, ·]k ≤ δkvk for all z ∈ Z and v ∈ TzZ;

(K) for every point in Z there exists a chart ϕ : B1(0) ⊆ Rd → Z around that

point such that k∂Pz

∂ξik, k

∂Qz

∂ξik ≤ ˜C, where (ξ1, . . . , ξd) are the coordinates

induced by ϕ.

Then there exists δ0 depending on C and ω such that, if δ < δ0, for every z ∈ Z

there exists w(z) ∈ Wz such that dI(z + w(z)) = 0 on Wz.

Moreover, we can choose w(z) such that kw(z)k = O(δ) and ∂w(z) ∂ξi = o(1) as

δ goes to 0, where O and o depend only on C and ω.

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3.6. Lyapunov-Schmidt reduction

Proof. Denote r > 0 a number so that ω(r) ≤ _4C1 and assume that δ < δ0 with

Cδ0 ≤ r₂. Fix z ∈ Z and define F : Wz → Wz∗as the function F (w) = dI(z +w)|Wz.

Notice that dI(z + w)[w1] = F (w)[w1] for every w, w1 ∈ W and consequently

it holds that

d2I(z + w)[w1, w2] = dF (w)[w1][w2] (3.6.1)

for every w, w1, w2 ∈ W . Then we have that

• kF (0)k ≤ kdI(z)k ≤ δ;

• thanks to (3.6.1) and (N D), we obtain that dF (0) is invertible (for example thanks to Hahn-Banach theorem) with k(dF (0))−1k ≤ C;

• thanks to (3.6.1) and (B1), we have instead that kdF (w)−dF (0)k = kd2I(z+

w) − d2_{I(z)k ≤ ω(kwk) for all w ∈ W} z.

Therefore F fulfills all the hypotheses required by Lemma 3.6.1 and thus we get that there exists w(z) ∈ Wz such that dI(z + w(z)) = 0 on Wz and kw(z)k ≤ 2Cδ.

In particular we have showed that kw(z)k = O(δ).

We know want to obtain the estimates on the derivatives of w(z). For this purpose, let us consider a chart ϕ : Bd

1(0) ⊆ Rd → Z as in (K) with induced

coordinates ξ = (ξ1_{, . . . , ξ}d_).

Since Qzw(z) = 0 for all z ∈ Z by definition of w(z), differentiating with

respect to ξi _{we obtain} ∂ ∂ξiQz ! w(z) + Qz ∂w(z) ∂ξi = 0 and consequently Qz ∂w(z) ∂ξi ≤ ˜Ckw(z)k ≤ 2 ˜CCδ .

Moreover it holds that dI(z + w(z)) ◦ Pz = 0, hence differentiating with respect

to ξi _{we obtain} dI(z + w(z)) ◦∂Pz ∂ξi + d 2_{I(z + w(z))} " ∂z ∂ξi + ∂w(z) ∂ξi # ◦ Pz = 0 ,

therefore we can estimate Pz∂w(z)_∂ξi as

Pz ∂w(z) ∂ξi ≤ C−1 d2I(z) " Pz ∂w(z) ∂ξi # ◦ Pz ≤ C−1 d2I(z + w(z)) " Pz ∂w(z) ∂ξi # ◦ Pz + C−1ω(kw(z)k) Pz ∂w(z) ∂ξi

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where the first term is estimated by d2I(z + w(z)) " Pz ∂w(z) ∂ξi # ◦ Pz ≤ ≤ dI(z + w(z)) ◦∂Pz ∂ξi + d2I(z + w(z)) " ∂z ∂ξi + Qz ∂w(z) ∂ξi # ◦ Pz ≤ ˜CkdI(z + w(z))k + d2I(z) " ∂z ∂ξi + Qz ∂w(z) ∂ξi # + + ω(kw(z)k) ∂z ∂ξi + Qz ∂w(z) ∂ξi Using (B2), it is immediate to obtain that

d2I(z) " ∂z ∂ξi + Qz ∂w(z) ∂ξi # ≤ δ ∂z ∂ξi + Qz ∂w(z) ∂ξi , moreover observe that

dI(z + w(z)) = dI(z + w(z)) ◦ Qz = dI(z) ◦ Qz+

ˆ 1 0

d2I(z + sw(z))[w(z)] ◦ Qzds .

However we know that kdI(z) ◦ Qzk ≤ kdI(z)k ≤ δ and that

kd2_{I(z + sw(z))[w(z)] ◦ Q}

zk ≤ kd2I(z)[w(z)] ◦ Qzk + ω(kw(z)k)kw(z)k

≤ (δ + ω(kw(z)k))kω(z)k , where we have used again (B1) and (B2).

Putting together all the inequalities, we thus obtain (1 − C−1ω(kw(z)k) ∂w(z) ∂ξi ≤ 2 ˜CCδ + C−1 " (δ + kw(z)k) ∂z ∂ξi + + (δ + kw(z)k) Qz ∂w(z) ∂ξi + + ˜C(δ + δkw(z)k + ω(kw(z)k))kw(z)k # , which implies easily that

∂w(z) ∂ξi tends to zero as δ → 0.

Finally, let us formalize that the set found in Proposition 3.6.2 is really a natural constraint for I.

Proposition 3.6.3. Let X, I, Z as in Proposition 3.6.2 and define φ : Z → R by

φ(z) = I(z + w(z)). If δ in the hypotheses of Proposition 3.6.2 is sufficiently small and if ¯z is a critical point for φ, then the point ¯z + w(¯z) is critical for I.

Construction of alpha-harmonic maps between spheres

Universit`

a degli Studi di Pisa

Construction of α-harmonic maps

between spheres

Contents

Introduction

Notations

1

Harmonic maps

1.1

Definition

1.2

Euler-Lagrange equation

1.3

Examples

1.4

Harmonic maps from surfaces

1.5

Regularity

1.6

Existence results

1.7

Harmonic maps from the sphere

2

Perturbed functional

2.1

Motivations

2.2

Definition

2.3

Compactness

2.4

Convergence

2.5

First variation

2.6

Second variation

3

Problem and tools

3.1

Harmonic maps between spheres

3.2

Dilations

3.3

Representation in polar coordinates

3.4

Statement

3.5

Idea of the proof

Find a constrained critical point

From the constrained critical point to the α-harmonic map

3.6

Lyapunov-Schmidt reduction